DOCSLIB.ORG

Mutually Unbiased Bases Are Complex Projective 2-Designs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mutually Unbiased Bases Are Complex Projective 2-Designs Mutually Unbiased Bases are Complex Projective 2-Designs Andreas Klappenecker Martin R¨otteler Department of Computer Science NEC Labs America, Inc. Texas A&M University 4 Independence Way College Station, TX, 77843–3112, USA Princeton, NJ 08540 U.S.A. Email: [email protected] Email: [email protected] Abstract— Mutually unbiased bases (MUBs) are a primitive that we want to determine the density matrix ρ of an ensemble used in quantum information processing to capture the principle of quantum systems using as few non-degenerate observables of complementarity. While constructions of maximal sets of d+1 as possible. We assume that it is possible to make a com- such bases are known for system of prime power dimension d, itis plete measurement of each observable O = x b b , unknown whether this bound can be achieved for any non-prime b∈B b | ih | power dimension. In this paper we demonstrate that maximal sets meaning that the statistics tr(ρ b b )= b ρ Pb is known for | ih | h | | i of MUBs come with a rich combinatorial structure by showing each eigenvalue xb in the spectral decomposition. Ivanovi´c that they actually are the same objects as the complex projective showed in [12] that complete measurements of at least d +1 2-designs with angle set {0, 1/d}. We also give a new and observables are needed to reconstruct the density matrix. He simple proof that symmetric informationally complete POVMs are complex projective 2-designs with angle set {1/(d+1)}. also showed that this lower bound is attained when d+1 non- degenerate pairwise complementary observables are used. I. INTRODUCTION A simple example is provided by the Pauli spin matrices σx, Two quantum mechanical observables are called comple- σy, σz. A complete measurement of these three observables mentary if and only if precise knowledge of one of them allows to reconstruct a 2 2 density matrix, a fact appar- implies that all possible outcomes are equally probable when ently known to Schwinger× [18]. Nowadays, we know how to measuring the other, see for example [19, p. 561]. The do this state tomography process—at least in principle—in principle of complementarity was introduced by Bohr [6] in dimensions d = 3, 4, and 5. It is an open problem whether 1928, and it had a profound impact on the further development it is possible to perform this kind of state tomography in of quantum mechanics. A recent application is the quantum dimension 6, because the construction of a set of 7 mutually key exchange protocol by Bennett and Brassard [3] that unbiased bases in dimension d =6 is elusive. exploits complementarity to secure the key exchange against eavesdropping. II. MUTUALLY UNBIASED BASES We mention a simple mathematical consequence of this d complementarity principle, which motivates some key notion. Definition 1: Two orthonormal bases B and C of C are ′ called mutually unbiased iff b c 2 =1/d holds for all b B Suppose that O and O are two hermitian d d matrices |h | i| ∈ representing a pair of complementary observables.× We assume and c C. arXiv:quant-ph/0502031v2 11 Feb 2005 ∈ that the eigenvalues of both matrices are multiplicity free. The goal is to construct d + 1 mutually unbiased bases It follows that the observables O and O′ respectively have (MUBs) in any dimension d 2. There are several con- ≥ orthonormal eigenbases B and B′ with basis vectors uniquely structions known to obtain MUBs. At least for prime power determined up to a scalar factor. dimension the problem is completely solved. This follows The complementarity of O and O′ implies that if a quantum from Constructions I-III below. However, in any dimension system is prepared in an eigenstate b′ of the observable O′, other than a prime power it is unknown if a maximal set and O is subsequently measured, then the probability to find of d + 1 MUBs can be found. The best known result is the system after the measurement in the state b B is given Construction IV below which only works in dimensions d by b b′ 2 =1/d. Recall that two orthonormal∈ bases B and which are squares and never gives a maximal set of MUBs. ′ |h |Ci|d B of are said to be mutually unbiased precisely when Construction I (Wootters and Fields [24]) Let q be an odd ′ 2 ′ ′ b b = 1/d holds for all b B and b B . Thus prime power. Define the|h | eigenbasesi| of non-degenerate complementary∈ ∈ observables 2 are mutually unbiased. Conversely, we can associate to a −1/2 tr(ax +bx) Cq va,b = q (ωp )x∈Fq , pair of mutually unbiased bases a pair of non-degenerate | i ∈ complementary observables. with ωp = exp(2πi/p). Then the standard basis together with There is a fundamental property of mutually unbiased bases the bases Ba = va,b b Fq , a Fq, form a set of q +1 that is invaluable in quantum information processing. Suppose mutually unbiased{| basesi | of∈ Cq}. ∈ Construction II (Galois Rings [13]) Let GR(4,n) be a finite Then we have that: Galois ring with Teichm¨uller set n. Define • M(pr)= pr +1 for p prime, r N, T ∈ 2πi • M(n) n +1 for all n N, v =2−n/2 exp tr(a +2b)x . ≤ ∈ N a,b 4 • M(mn) min M(m),M(n) for all m,n . | i x∈Tn ≥ { } ∈ • M(d2) N(d), where N(d) is the number of mutually ≥ Then the standard basis together with the bases Ma = orthogonal Latin squares of size d d. n va,b b n , a n, form a set of 2 + 1 mutually × n An open problem is to show that lim inf n→∞ M(n)= . unbiased{| i | bases∈ T of} C2∈. T ∞ Construction III (Bandyopadhyay et al. [1]) Suppose there III. WELCH’S LOWER BOUNDS exist subsets 1,..., m of a unitary error basis such that Suppose that X is a finite nonempty set of vectors of unit C C B Cd i = d, i j = 1d for i = j, and the elements of i norm in the complex vector space . The vectors in X satisfy pairwise|C | commute.C ∩C Let{ M} be a matrix6 which diagonalizes C. the inequalities i Ci Then M1,...,Mm are MUBs. 1 1 x y 2k , (1) Construction IV (Wocjan and Beth [23]) Suppose there are X 2 |h | i| ≥ d+k−1 | | x,yX∈X k w mutually orthogonal Latin squares [4], each of size d d over the symbol set S = 1,...,d . Then w + 2 MUBs× for all integers k 0. Welch derived these bounds in [22] ≥ in dimension d2 can be constructed{ } as follows. With each to obtain a lower bound on the maximal cross-correlation of Latin square L (and additionally the square (1, 2,...,n)t spreading sequences of synchronous code-division multiple- (1,..., 1)) we can associate vectors of length d over the⊗ access systems. Blichfeld [5] and Sidelnikov [20] derived alphabet 1,...,d2 : for each symbol α S define a vector similar bounds for real vectors of unit norm. {d } ∈ sL,α C as follows: start with the empty list sL,α = . A set X attaining the Welch bound (1) for k = 1 is Then∈ traverse the elements of L column-wise starting at the∅ called a WBE-sequence set, a notion popularized by Massey upper left corner. Whenever α occurs in position (i, j) in L, and Mittelholzer [15] and others. Using equation (1), it is then append the number i + jd to the list sL,α. The other straightforward to check that the union of d + 1 mutually ingredient to construct these MUBs is an arbitrary complex unbiased bases of Cd form a WBE-sequence set. These Hadamard matrix H = (hi,j ) of size d d. For each Latin extremal sets of mutually unbiased bases are even better, since square L and each α, j 1,...,d define× a normalized they also attain the Welch bound for k = 2. In fact, we √ ∈d { } show that a sequence set attains the Welch bounds (1) for all vector vL,α,j := 1/ d i=1 esL,α[i]hi,j , where ei are the | i 2 elementary basis vectorsP in Cd . Then the bases given by k t if and only if it is a t-design in the complex projective space≤ CP d−1. BL := vL,α,j : α, j = 1,...,d together with the identity {| i } Let us introduce some notation. Let Sd−1 denote the sphere matrix 1d2 form a set of w +2 MUBs. of unit vectors in the complex vector space Cd. We say that Example 1: In dimension d = 3 Construction I yields the two vectors u and v of Sd−1 are equivalent, in signs u bases v, if and only if u = eiθv for some θ R. It is easy to≡ −1/2 2 2 ∈ 3 (1, 1, 1), (1,ω3,ω3), (1,ω3,ω3) , see that is an equivalence relation. We denote the quotient { } ≡ − − − −1/2 2 2 manifold Sd 1/ by CSd 1. Notice that the manifold CSd 1 3 (1,ω3,ω3), (1,ω3, 1), (1, 1,ω3) , { } is isomorphic to≡ the complex projective space CP d−1, but we 3−1/2 (1,ω2,ω2), (1,ω , 1), (1, 1,ω ) , { 3 3 3 3 } prefer the former notation because normalizing vectors to unit which together with the standard basis 13 form a maximal length is common practice in quantum computing.
Load more

Recommended publications
  • Arxiv:1001.0543V2 [Quant-Ph] 10 Oct 2010
    Optimal reconstruction of the states in qutrits system Fei Yan,1 Ming Yang∗,1 and Zhuo-Liang Cao2 1Key Laboratory of Opto-electronic Information Acquisition and Manipulation, Ministry of Education, School of Physics and Material Science, Anhui University, Hefei 230039, People's Republic of China 2Department of Physics & Electronic Engineering, Hefei Normal University, Hefei 230061, People's Republic of China Based on mutually unbiased measurements, an optimal tomographic scheme for the multiqutrit states is pre- sented explicitly. Because the reconstruction process of states based on mutually unbiased states is free of information waste, we refer to our scheme as the optimal scheme. By optimal we mean that the number of the required conditional operations reaches the minimum in this tomographic scheme for the states of qutrit systems. Special attention will be paid to how those different mutually unbiased measurements are realized; that is, how to decompose each transformation that connects each mutually unbiased basis with the standard computational basis. It is found that all those transformations can be decomposed into several basic implementable single- and two-qutrit unitary operations. For the three-qutrit system, there exist five different mutually unbiased-bases structures with different entanglement properties, so we introduce the concept of physical complexity to min- imize the number of nonlocal operations needed over the five different structures. This scheme is helpful for experimental scientists to realize the most economical reconstruction of quantum states in qutrit systems. PACS numbers: 03.65.Wj, 03.65.Ta, 03.65.Ud, 03.67.Mn I. INTRODUCTION is redundancy in these results in the previously used quantum tomography processes [12], which causes a resources waste.
    [Show full text]
  • Mutually Unbiased Bases and Symmetric Informationally Complete Measurements in Bell Experiments
    Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments Armin Tavakoli,1 Máté Farkas,2, 3 Denis Rosset,4 Jean-Daniel Bancal,1 and J˛edrzejKaniewski5 1Department of Applied Physics, University of Geneva, 1211 Geneva, Switzerland 2Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre, Faculty of Mathematics, Physics and Informatics, University of Gdansk, 80-952 Gdansk, Poland 3International Centre for Theory of Quantum Technologies, University of Gdansk, 80-308 Gdansk, Poland 4Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada 5Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland (Dated: October 21, 2020) Mutually unbiased bases (MUBs) and symmetric informationally complete projectors (SICs) are crucial to many conceptual and practical aspects of quantum theory. Here, we develop their role in quantum nonlocality by: i) introducing families of Bell inequalities that are maximally violated by d-dimensional MUBs and SICs respectively, ii) proving device-independent certification of natural operational notions of MUBs and SICs, and iii) using MUBs and SICs to develop optimal-rate and nearly optimal-rate protocols for device independent quantum key distribution and device-independent quantum random number generation respectively. Moreover, we also present the first example of an extremal point of the quantum set of correlations which admits physically inequivalent quantum realisations. Our results elaborately demonstrate the foundational and practical relevance of the two most important discrete Hilbert space structures to the field of quantum nonlocality. One-sentence summary — Quantum nonlocality is de- a single measurement. Since MUBs and SICs are both con- veloped based on the two most celebrated discrete struc- ceptually natural, elegant and (as it turns out) practically im- tures in quantum theory.
    [Show full text]
  • Morphophoric Povms, Generalised Qplexes, Ewline and 2-Designs
    Morphophoric POVMs, generalised qplexes, and 2-designs Wojciech S lomczynski´ and Anna Szymusiak Institute of Mathematics, Jagiellonian University,Lojasiewicza 6, 30-348 Krak´ow, Poland We study the class of quantum measurements with the property that the image of the set of quantum states under the measurement map transforming states into prob- ability distributions is similar to this set and call such measurements morphophoric. This leads to the generalisation of the notion of a qplex, where SIC-POVMs are re- placed by the elements of the much larger class of morphophoric POVMs, containing in particular 2-design (rank-1 and equal-trace) POVMs. The intrinsic geometry of a generalised qplex is the same as that of the set of quantum states, so we explore its external geometry, investigating, inter alia, the algebraic and geometric form of the inner (basis) and the outer (primal) polytopes between which the generalised qplex is sandwiched. In particular, we examine generalised qplexes generated by MUB-like 2-design POVMs utilising their graph-theoretical properties. Moreover, we show how to extend the primal equation of QBism designed for SIC-POVMs to the morphophoric case. 1 Introduction and preliminaries Over the last ten years in a series of papers [2,3, 30{32, 49, 68] Fuchs, Schack, Appleby, Stacey, Zhu and others have introduced first the idea of QBism (formerly quantum Bayesianism) and then its probabilistic embodiment - the (Hilbert) qplex, both based on the notion of symmetric informationally complete positive operator-valued measure
    [Show full text]
  • Five Isoentangled Mutually Unbiased Bases and Mixed States Designs Praca Licencjacka Na Kierunku fizyka Do´Swiadczalna I Teoretyczna
    UNIWERSYTET JAGIELLONSKIW´ KRAKOWIE WYDZIAŁ FIZYKI,ASTRONOMII I INFORMATYKI STOSOWANEJ Jakub Czartowski Nr albumu: 1125510 Five isoentangled mutually unbiased bases and mixed states designs Praca licencjacka Na kierunku fizyka do´swiadczalna i teoretyczna Praca wykonana pod kierunkiem Prof. Karola Zyczkowskiego˙ Instytut Fizyki Kraków, 24 maja 2018 JAGIELLONIAN UNIVERSITYIN CRACOW FACULTY OF PHYSICS,ASTRONOMY AND APPLIED COMPUTER SCIENCE Jakub Czartowski Student number: 1125510 Five isoentangled mutually unbiased bases and mixed states designs Bachelor thesis on a programme Fizyka do´swiadczalnai teoretyczna Prepared under supervision of Prof. Karol Zyczkowski˙ Institute of Physics Kraków, 24 May 2018 Contents Abstract 5 Acknowledgements5 1 Introduction6 2 Space of pure quantum states7 3 Space of mixed quantum states7 4 Complex projective t-designs9 4.1 Symmetric Informationally Complete (SIC) POVM . 10 4.2 Mutually Unbiased Bases (MUB) . 10 5 Mixed states designs 10 5.1 Mean traces on the space of mixed states . 12 6 Notable examples 12 6.1 t = 1 - Positive Operator Valued Measure . 12 6.2 N=2, t=2, M=8 - Isoentangled SIC POVM for d 4................. 13 Æ 6.3 Short interlude - Numerical search for isoentangled MUB . 13 6.4 N=2, t=2, M=20 - Isoentangled MUB . 15 6.5 N=2, t=2, M=10 - The standard set of 5 MUB in H4 ................ 16 6.6 N=2, t=3, M=10 - Hoggar example number 24 . 17 6.7 N=2, t=3, M=20(40) - Witting polytope . 18 6.8 Interlude - Summary for N 2 ............................ 18 Æ 6.9 N=3, t=2, M=90 - MUB . 19 6.9.1 Standard representation .
    [Show full text]
  • Arxiv:1809.09986V2 [Physics.Optics] 30 Apr 2019
    Using all transverse degrees of freedom in quantum communications based on a generic mode sorter YIYU ZHOU,1,9,* MOHAMMAD MIRHOSSEINI,1,2,9 STONE OLIVER,1,3 JIAPENG ZHAO,1 SEYED MOHAMMAD HASHEMI RAFSANJANI,1,4 MARTIN P. J. LAVERY,5 ALAN E.WILLNER,6 AND ROBERT W. BOYD1,7,8 1The Institute of Optics, University of Rochester, Rochester, New York 14627, USA 2Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA 3Department of Physics, Miami University, Oxford, Ohio 45056, USA 4Department of Physics, University of Miami, Coral Gables, Florida 33146, USA 5School of Engineering, University of Glasgow, Glasgow, Scotland G12 8LT, UK 6Department of Electrical Engineering, University of Southern California, Los Angeles, California 90089, USA 7Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada [email protected] 9These authors contributed equally *[email protected] Abstract: The dimension of the state space for information encoding offered by the transverse structure of light is usually limited by the finite size of apertures. The widely used orbital angular momentum (OAM) number of Laguerre-Gaussian (LG) modes in free-space communications cannot achieve the theoretical maximum transmission capacity unless the radial degree of freedom is multiplexed into the protocol. While the methodology to sort the radial quantum number has been developed, the application of radial modes in quantum communications requires an additional ability to efficiently measure the superposition of LG modes in the mutually unbiased basis. Here we develop and implement a generic mode sorter that is capable of sorting the superposition of LG modes through the use of a mode converter.
    [Show full text]
  • Arxiv:1704.04609V3 [Quant-Ph] 5 Jul 2017 Lations [5, 6]
    Quantum measurements with prescribed symmetry Wojciech Bruzda Institute of Physics, Jagiellonian University, Krak´ow,Poland Dardo Goyeneche Institute of Physics, Jagiellonian University, Krak´ow,Poland and Faculty of Applied Physics and Mathematics, Technical University of Gda´nsk,80-233 Gda´nsk,Poland Karol Zyczkowski_ Institute of Physics, Jagiellonian University, Krak´ow,Poland and Center for Theoretical Physics, Polish Academy of Sciences, Warsaw, Poland (Dated: July 4, 2017) We introduce a method to determine whether a given generalised quantum measurement is iso- lated or it belongs to a family of measurements having the same prescribed symmetry. The technique proposed reduces to solving a linear system of equations in some relevant cases. As consequence, we provide a simple derivation of the maximal family of Symmetric Informationally Complete measure- ment (SIC)-POVM in dimension 3. Furthermore, we show that the following remarkable geometrical structures are isolated, so that free parameters cannot be introduced: (a) maximal sets of mutually unbiased bases in prime power dimensions from 4 to 16, (b) SIC-POVM in dimensions from 4 to 16 and (c) contextuality Kochen-Specker sets in dimension 3, 4 and 6, composed of 13, 18 and 21 vectors, respectively. Keywords: Mutually unbiased bases, SIC- for different convenient purposes. For example, POVM, defect of a unitary matrix. from the one-parametric family of SIC-POVM ex- isting in dimension three [1] only a single member maximizes the informational power, that is, the classical capacity of a quantum-classical channel I. INTRODUCTION generated by the SIC-POVM [14]. Furthermore, inequivalent sets of MUB provide different estima- Positive Operator Valued Measure (POVM) is tion of errors in quantum tomography [15].
    [Show full text]
  • Uncertainty Equality with Quantum Memory and Its Experimental Verification
    www.nature.com/npjqi ARTICLE OPEN Uncertainty equality with quantum memory and its experimental verification Hengyan Wang1,2,3,4, Zhihao Ma5, Shengjun Wu6, Wenqiang Zheng7, Zhu Cao8, Zhihua Chen7, Zhaokai Li1,3,4, Shao-Ming Fei9,10, Xinhua Peng 1,3,4, Vlatko Vedral11,12 and Jiangfeng Du1,3,4 As a very fundamental principle in quantum physics, uncertainty principle has been studied intensively via various uncertainty inequalities. A natural and fundamental question is whether an equality exists for the uncertainty principle. Here we derive an entropic uncertainty equality relation for a bipartite system consisting of a quantum system and a coupled quantum memory, based on the information measure introduced by Brukner and Zeilinger (Phys. Rev. Lett. 83:3354, 1999). The equality indicates that the sum of measurement uncertainties over a complete set of mutually unbiased bases on a subsystem is equal to a total, fixed uncertainty determined by the initial bipartite state. For the special case where the system and the memory are the maximally entangled, all of the uncertainties related to each mutually unbiased base measurement are zero, which is substantially different from the uncertainty inequality relation. The results are meaningful for fundamental reasons and give rise to operational applications such as in quantum random number generation and quantum guessing games. Moreover, we experimentally verify the measurement uncertainty relation in the presence of quantum memory on a five-qubit spin system by directly measuring the corresponding quantum mechanical observables, rather than quantum state tomography in all the previous experiments of testing entropic uncertainty relations. npj Quantum Information (2019) 5:39 ; https://doi.org/10.1038/s41534-019-0153-z INTRODUCTION Shannon entropies of the measurement results, the uncertainty 1 4 fi The uncertainty principle is one of the most important principle in principle can be formulated as Hθ þ Hτ log2 c.
    [Show full text]
  • New Bounds for Mutually Unbiased Maximally Entangled Bases in C D
    New bounds for Mutually unbiased maximally entangled bases in Cd ⊗ Ckd XiaoyaCheng1 ∗ YunShang1 2 y 1 Institute of Mathematics, AMSS, CAS, Beijing 100190, P.R.China 2NCMIS, AMSS, CAS, Beijing 100190, P.R.China Abstract. Mutually unbiased bases which is also maximally entangled bases is called mutually unbiased maximally entangled bases (MUMEBs). We study the construction of MUMEBs in bipartite system. In detail, we construct 2(pa − 1) MUMEBs in Cd ⊗ Cd by properties of Gauss sums for arbitrary odd d. It improves the known lower bound pa − 1 for odd d. Furthermore, we construct MUMEBs in Cd ⊗ Ckd for general k ≥ 2 and odd d. We get the similar lower bounds as k; b are both single prime powers. Particularly, when k is a square number, by using mutually orthogonal Latin squares, we can construct more MUMEBs in Cd ⊗ Ckd. Keywords: mutually unbiased bases, maximally entangled states, Pauli matrices, mutually orthogonal Latin squares 1 Introduction etc., all of which depend on an explicit set of maximal- 0 ly entangled states. A basis B of d ⊗ d is called a The notion of mutually unbiased bases was first intro- C C maximally entangled basis (MEB) if it consists of dd0 duced by Schwinger [1] in 1960. Two orthogonal bases d d d maximally entangled states. Let A = fB1; B2;:::; Bmg B1 = fjφiigi=1 and B2 = fj jigj=1 of C are called d d0 be a set of orthonormal MEBs in C ⊗ C . We call A mutually unbiased if jhφ j ij = p1 ; (1 ≤ i; j ≤ d): I- i j d a set of mutually unbiased maximally entangled bases vanovic found the first application for mutually unbiased (MUMEBs) if every pair in fB1; B2;:::; Bmg is mutually bases (MUBs) in the problem of quantum state determi- unbiased.
    [Show full text]
  • SIC-Povms and Compatibility Among Quantum States
    mathematics Article SIC-POVMs and Compatibility among Quantum States Blake C. Stacey Physics Department, University of Massachusetts Boston, 100 Morrissey Boulevard, Boston, MA 02125, USA; [email protected]; Tel.: +1-617-287-6050; Fax: +1-617-287-6053 Academic Editors: Paul Busch, Takayuki Miyadera and Teiko Heinosaari Received: 1 March 2016; Accepted: 14 May 2016; Published: 1 June 2016 Abstract: An unexpected connection exists between compatibility criteria for quantum states and Symmetric Informationally Complete quantum measurements (SIC-POVMs). Beginning with Caves, Fuchs and Schack’s "Conditions for compatibility of quantum state assignments", I show that a qutrit SIC-POVM studied in other contexts enjoys additional interesting properties. Compatibility criteria provide a new way to understand the relationship between SIC-POVMs and mutually unbiased bases, as calculations in the SIC representation of quantum states make clear. This, in turn, illuminates the resources necessary for magic-state quantum computation, and why hidden-variable models fail to capture the vitality of quantum mechanics. Keywords: SIC-POVM; qutrit; post-Peierls compatibility; Quantum Bayesian; QBism PACS: 03.65.Aa, 03.65.Ta, 03.67.-a 1. A Compatibility Criterion for Quantum States This article presents an unforeseen connection between two subjects originally studied for separate reasons by the Quantum Bayesians, or to use the more recent and specific term, QBists [1,2]. One of these topics originates in the paper “Conditions for compatibility of quantum state assignments” [3] by Caves, Fuchs and Schack (CFS). Refining CFS’s treatment reveals an unexpected link between the concept of compatibility and other constructions of quantum information theory.
    [Show full text]
  • Mutually Unbiased Bases and Trinary Operator Sets for N Qutrits
    Dartmouth College Dartmouth Digital Commons Open Dartmouth: Published works by Dartmouth faculty Faculty Work 7-2-2004 Mutually Unbiased Bases and Trinary Operator Sets for N Qutrits Jay Lawrence Dartmouth College Follow this and additional works at: https://digitalcommons.dartmouth.edu/facoa Part of the Physics Commons Dartmouth Digital Commons Citation Lawrence, Jay, "Mutually Unbiased Bases and Trinary Operator Sets for N Qutrits" (2004). Open Dartmouth: Published works by Dartmouth faculty. 2961. https://digitalcommons.dartmouth.edu/facoa/2961 This Article is brought to you for free and open access by the Faculty Work at Dartmouth Digital Commons. It has been accepted for inclusion in Open Dartmouth: Published works by Dartmouth faculty by an authorized administrator of Dartmouth Digital Commons. For more information, please contact [email protected]. PHYSICAL REVIEW A 70, 012302 (2004) Mutually unbiased bases and trinary operator sets for N qutrits Jay Lawrence Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA (Received 13 March 2004; published 2 July 2004) A compete orthonormal basis of N-qutrit unitary operators drawn from the Pauli group consists of the identity and 9N −1 traceless operators. The traceless ones partition into 3N +1 maximally commuting subsets (MCS’s) of 3N −1 operators each, whose joint eigenbases are mutually unbiased. We prove that Pauli factor groups of order 3N are isomorphic to all MCS’s and show how this result applies in specific cases. For two qutrits, the 80 traceless operators partition into 10 MCS’s. We prove that 4 of the corresponding basis sets must be separable, while 6 must be totally entangled (and Bell-like).
    [Show full text]
  • Quantum Communication with Photons
    Quantum communication with photons Mario Krenn1,2,3, Mehul Malik1,2, Thomas Scheidl1,2, Rupert Ursin1,2, and Anton Zeilinger1,2,3 1Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria 2Vienna Center for Quantum Science and Technology, Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria 3correspondence to: [email protected] and [email protected] January 5, 2017 Abstract The secure communication of information plays an ever increasing role in our society today. Classical methods of encryption inherently rely on the difficulty of solving a problem such as finding prime factors of large numbers and can, in principle, be cracked by a fast enough machine. The burgeoning field of quantum communication relies on the fundamental laws of physics to offer unconditional information security. Here we in- troduce the key concepts of quantum superposition and entanglement as well as the no-cloning theorem that form the basis of this field. Then, we review basic quantum communication schemes with single and entangled photons and discuss recent experimental progress in ground and space- based quantum communication. Finally, we discuss the emerging field of high-dimensional quantum communication, which promises increased data rates and higher levels of security than ever before. We discuss re- cent experiments that use the orbital angular momentum of photons for sharing large amounts of information in a secure fashion. arXiv:1701.00989v1 [quant-ph] 4 Jan 2017 1 Contents 1 Introduction 3 1.1 The Quantum Bit . 3 1.2 Entanglement . 5 1.3 Mutually Unbiased Bases .
    [Show full text]
  • Equiangular Vectors Approach to Mutually Unbiased Bases
    Entropy 2013, 15, 1726-1737; doi:10.3390/e15051726 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.com/journal/entropy Article Equiangular Vectors Approach to Mutually Unbiased Bases Maurice R. Kibler 1;2;3 1 Faculte´ des Sciences et Technologies, Universite´ de Lyon, 37 rue du repos, 69361 Lyon, France 2 Departement´ de Physique, Universite´ Claude Bernard Lyon 1, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne, France 3 Groupe Theorie,´ Institut de Physique Nucleaire,´ CNRS/IN2P3, 4 rue Enrico Fermi, 69622 Villeurbanne, France; E-Mail: [email protected]; Tel: 33(0)472448235, FAX: 33(0)472448004 Received: 26 April 2013 / Accepted: 6 May 2013 / Published: 8 May 2013 Abstract: Two orthonormal bases in the d-dimensional Hilbert space are said to be unbiased if the square modulus of the inner product of any vector of one basis with any vector of the 1 other equals d . The presence of a modulus in the problem of finding a set of mutually unbiased bases constitutes a source of complications from the numerical point of view. Therefore, we may ask the question: Is it possible to get rid of the modulus? After a short review of various constructions of mutually unbiased bases in Cd, we show how to transform the problem of finding d + 1 mutually unbiased bases in the d-dimensional space Cd (with a modulus for the inner product) into the one of finding d(d + 1) vectors in the d2-dimensional 2 2 space Cd (without a modulus for the inner product). The transformation from Cd to Cd corresponds to the passage from equiangular lines to equiangular vectors.
    [Show full text]